Understanding QFT vs QM: A Beginner's Guide to the Differences and Similarities

[waterfall]

Yes, in some sense we are dealing with harmonic oscillators, but these are quantum simple harmonic oscillators, not classical SHO. Just as a particle in nonrelativistic quantum mechanics, like a quantum harmonic oscillator, does not have a definite position but only a probability of being measured at different positions, in quantum field theory there is no definite position associated with field quanta AKA particles.

When you mentioned "field quanta", are you referring to operator field quanta or actual field quanta? This is because as detailed in message 29, the field in QFT are field operator, not the usual field we understood as electromagnetic field for example.

 

[Oudeis Eimi]

Are you saying not all physicists with Ph.D. are experts in QFT? I thought they all wer. But using Fock space noninteracting terms, how could they make the Large Hadron Collider function and successfully predict those scattering angles and interactions of the numerous particles. Is Fock space enough to analyze them including predicting the mass of the Higgs? Or do Large Hadron Collider physicists use purely rigorous QFT that normal physicists don't tackle?

Most physicists work in condensed matter, not particle or high-energy physics. They have some
knowledge of QFT (mostly the non-relativistic kind) as part of their training in QM, but needn't be
experts in the mathematical foundations of QM.

So how does one imagine a quantum field? I thought it should have particles vibrating like harmonic oscillator.. but now saying particles don't have position.. then how does one picture it? Or is it possible space and time only occur during interaction with the quantum field, and without interaction, space and time doesn't really exist as we know it in the quantum field? And it is just a blob of untime and unspace?

The particles are excitations, the most basic 'vibration states' of the fields. When we say they don't
necessarily have a position, what we mean, in layman's language, is that those 'basic vibrations'
aren't confined to a single point in space. Note however that they may (but don't NEED to be)
confined to a very tiny region from our macroscopic point of view. This is completely analogous to
the case of non-relativistic ordinary QM.

As to how to visualise a quantum field... well, quantum operators behave a lot like stochastic
variables. They have an expectation value and a complete set of moments which give you the
indeterminacy of said expectation value. So in principle, any such operator can be visualised as
a 'fuzzy' quantity, centered around the expectation value and with the fuzziness being proportional
to the indeterminacy. So for the case of a field, it's a 'fuzzy' field.

As a visualisation technique, this is probably only useful for bosonic fields in states such that
the indeterminacy is much smaller than the expectation value. This is the case for instance for
the electromagnetic field in most ordinary cases. Fermionic fields OTOH don't have a classic
limit and are thus much harder to visualise.

 

[Oudeis Eimi]

Wikipedia entry on QFT is wrong then, it depicts things as almost complete and rosy. For example the following words are not right:

http://en.wikipedia.org/wiki/Quantum_field_theory

Wiki:"Quantum field theory is thought by many[who?] to be the unique and correct outcome of combining the Rules of Quantum Mechanics with special relativity."

Fact: it is not exactly correct as you emphasized.

Wiki:"In perturbative quantum field theory, the forces between particles are mediated by other particles. The electromagnetic force between two electrons is caused by an exchange of photons. Intermediate vector bosons mediate the weak force and gluons mediate the strong force. "

Fact: Fock space doesn't handle interactions so those pertubative approach are just temporary and is fundamentally invalid"

Wiki:"In QFT, photons are not thought of as "little billiard balls" but are rather viewed as field quanta – necessarily chunked ripples in a field, or "excitations", that "look like" particles."

Fact: Particles don't have positions so they are not really excitations of the field. One must not visualize it that way.

Agree with everything? Maybe its time to correct Wiki and state things are not that rosy and indeed bleak.

No, not agreed. Perturbative techniques work well within their range of applicability. They're not the
ideal solution, but are necessary for those cases where the full solution to the problem isn't available.
Note such techniques are /extensively/ used across both pure and applied physics (including
engineering). For instance, we don't have a general solution for the N-body problem, so we need
to resort to approximations like numerical and/or perturbative methods.

Fact: Particles don't have positions so they are not really excitations of the field. One must not visualize it that way.

This doesn't follow. Your conclusion is invalid.

 

[Oudeis Eimi]

When you mentioned "field quanta", are you referring to operator field quanta or actual field quanta? This is because as detailed in message 29, the field in QFT are field operator, not the usual field we understood as electromagnetic field for example.

Waterfall, a quantum field is a quantum 'quantity'. In the formalism of quantum physics, these are
operators (or POVMs, which are a related but more complicated object). The 'actual' field IS the
'operator' field.

I'll give you two examples: the total momentum of a system, P, and the electromagnetic field, A.
In CLASSICAL physics, these, or their components in some reference frame, are numbers.
P = {Px, Py, Pz}; A = {phi, Ax, Ay, Az}.

In QUANTUM physics, these are operators. That's a more complicated kind of object. An important
difference with the above case is, operators don't have a value by themselves. This is where the
state comes in in the theory. Quantum states give operators their values (and their indeterminacy).

So, while in classical physics you have A=A(x,y,z,t) as a vector with a definite value assigned
to every point (x,y,z), in quantum physics you have A=A(x,y,z,t) as an operator field, that is,
an operator assigned to every point of space (and time). Once you're given a state you can
assign a value (actually, an expectation value and an indeterminacy) to those operators. If the
indeterminacy is sufficiently small, it can be ignored and you recover the classical field (this
can only happen for fields which do possesses a classical limit, of course. The em field does.)

 

[mysearch]

A quantum field is a quantum 'quantity'. In the formalism of quantum physics, these are operators (or POVMs, which are a related but more complicated object). The 'actual' field IS the 'operator' field.

Is the word ‘actual’ in the statement above based on mathematical consistency or any level of physical verification?

I do not want to be accused of blatant scepticism, although it is said that a certain amount is healthy. Equally, I do not want be accused of just cherry-picking comments by other people out of context just because they might appear to question some aspect of QFT. However, from the perspective of somebody simply interested in the subject, I am beginning to wonder just how many years of maths is now required to even come close to understanding QFT, let alone questioning any of its fundamental premises. As such, it seems that QFT may now extend beyond the reach of most people to quantify for themselves and therefore they must “stand on the shoulders of giants” or, at least, on the shoulders of somebody taller than themselves. However, it seems that any conclusions drawn will still depend on whose shoulders you decide to pick, e.g. see article “The search for a quantum field theory” for a somewhat pessimistic, and possibly outdated, take on the current state of play. Of course, this author, although apparently well qualified, may have simply lost his way and been left behind by leading edge thinking. Therefore, I am assuming that his concerns can now be dismissed?

Wikipedia entry on QFT is wrong then, it depicts things as almost complete and rosy……………
Maybe it’s time to correct Wiki and state things are not that rosy and indeed bleak.

...most of them have at least taken a QFT course. But that's not the point I was trying to make when I mentioned rigorous QFT. The point was questions like what the Hilbert space of the interacting theory is aren't answered in typical QFT courses, or typical QFT books. Actually, I don't think anyone even knows how to properly define the Hilbert space for QED in 3+1 dimensions. (Maybe they know that and are still struggling with other things, but they're struggling with something, because I know that no rigorous version of QED in 3+1 dimensions has been found). I suspect that even some QFT experts don't know rigorous QFT. It's like an entirely different field of physics. A typical student at an "introduction to QFT" course would probably need two more years of math before he can really begin to learn rigorous QFT.

 

[waterfall]

Most physicists work in condensed matter, not particle or high-energy physics. They have some
knowledge of QFT (mostly the non-relativistic kind) as part of their training in QM, but needn't be
experts in the mathematical foundations of QM.


The particles are excitations, the most basic 'vibration states' of the fields. When we say they don't
necessarily have a position, what we mean, in layman's language, is that those 'basic vibrations'
aren't confined to a single point in space. Note however that they may (but don't NEED to be)
confined to a very tiny region from our macroscopic point of view. This is completely analogous to
the case of non-relativistic ordinary QM.

As to how to visualise a quantum field... well, quantum operators behave a lot like stochastic
variables. They have an expectation value and a complete set of moments which give you the
indeterminacy of said expectation value. So in principle, any such operator can be visualised as
a 'fuzzy' quantity, centered around the expectation value and with the fuzziness being proportional
to the indeterminacy. So for the case of a field, it's a 'fuzzy' field.

As a visualisation technique, this is probably only useful for bosonic fields in states such that
the indeterminacy is much smaller than the expectation value. This is the case for instance for
the electromagnetic field in most ordinary cases. Fermionic fields OTOH don't have a classic
limit and are thus much harder to visualise.

Electromagnetic field and fermionic (or matter) fields are not directly the gauge fields which are unobservable. We can observe the electromagnetic field. Is it possible we just haven't yet invented the technology to detect matter fields? We detect electrons by scattering events and hits in detector. But the more subtle matter fields may need other methods of detection. What would it take to detect them?

 

[Oudeis Eimi]

Is the word ‘actual’ in the statement above based on mathematical consistency or any level of physical verification?

It was sloppy language on my part. What I meant is, the operator-valued fields are
the mathematical models that correspond in the quantum theory to the number-valued
fields of the classical theory.

 

[StevieTNZ]

I think it's believed that QED is fundamentally unsound - it is inconsistent at high energies. Strictly speaking, there's no proof of that since it's only perturbatively unsound.

Is this why formulating a Quantum Gravity theory is being problematic?

 

[lugita15]

Is this why formulating a Quantum Gravity theory is being problematic?

No, as far as we know quantum gravity may not even have a Landau pole, which is the big issue with QED.

It's hard to do exact calculations in any quantum field theory, so in order to get approximate answers we use perturbation theory to get infinite series. But it turns out that most of these series are divergent, so we apply a procedure known as renormalization to get finite results. Renormalization requires knowing the values of so-called "running constants", parameters which must be determined by experiment. Most theories like QED and QCD just require the determination of a few such constants, but quantum gravity requires infinitely many constants, and it's not very practical to do infinite experiments. The hope with theories like string theory is that perhaps there are undiscovered symmetries (e.g. supersymmetry) which would provide relations between these constants so that only finitely many experiments need to be done. Another idea is to somehow to quantum gravity calculations nonperturbatively, and thus avoid the need for renormalization altogether.

BTW, the Landau pole problem with QED is that at very high energies, renormalization fails to give sensible answers. Again, a possible solution to this would be to find a nonperturbative method of calculation.

 

[waterfall]

No, as far as we know quantum gravity may not even have a Landau pole, which is the big issue with QED.

It's hard to do exact calculations in any quantum field theory, so in order to get approximate answers we use perturbation theory to get infinite series. But it turns out that most of these series are divergent, so we apply a procedure known as renormalization to get finite results. Renormalization requires knowing the values of so-called "running constants", parameters which must be determined by experiment. Most theories like QED and QCD just require the determination of a few such constants, but quantum gravity requires infinitely many constants, and it's not very practical to do infinite experiments. The hope with theories like string theory is that perhaps there are undiscovered symmetries (e.g. supersymmetry) which would provide relations between these constants so that only finitely many experiments need to be done. Another idea is to somehow to quantum gravity calculations nonperturbatively, and thus avoid the need for renormalization altogether.

BTW, the Landau pole problem with QED is that at very high energies, renormalization fails to give sensible answers. Again, a possible solution to this would be to find a nonperturbative method of calculation.

After reading many books on Quantum Field Theory (each in one sitting). I got the feeling that somehow QFT is only an approach for calculational purposes. This means it is not something permanent. Meaning Quantum Field Theory can someday be replaced by others which doesn't involve the quantum field especially the matter field by second quantization. Do you agree with this analysis? Therefore when MySearch asked in the other thread "what is the field in QFT?". Well. I think the fields are just for certain calculational approach and is not something definite like spin and can be replaced someday. Do you agree?

The latest I'm reading is M.Y. Han's book "A Story Of Light: A Short Introduction To Quantum Field Theory Of Quarks And Leptons"

https://www.amazon.com/dp/9812560343/?tag=pfamazon01-20

Which part of the following do you think is inaccurate and why?

The first leap of faith is the introduction of the concept of matter fields, as discussed in Chapter 7. The quantization of the electromagentic field successfully incorporated photons as the quanta of that field and - this is critical - the electromagnetic field (the four-vector potential) satisfied a classical wave equation identical to the Klein-Gordon equation for zero-mass case. A classical wave equation of the 19th century turned out to be the same as the defining wave equation of relativistic quantum mechanics of the 20th century! This then led to the first leap of faith - the grandest emulation of radiation by matter - that all matter particles, electrons and positrons initially and now extended to all matter particles, quarks and leptons, should be considered as quanta of their own quantized fields, each to its own. The wavefunctions of the relativistic quantum mechanics morphed into classical fields. This conceptual transition from relativistic quantum mechanical wavefunctions to classical fields was the first necesary step toward quantized matter fields. Whether such emulation of radiation by matter is totally justifiable remains an open question. It will remain an open question until we successfully achive completely satisfactory quantum field theory of matter, a goal not yet fully achieved.

 

[Fredrik]

After reading many books on Quantum Field Theory (each in one sitting). I got the feeling that somehow QFT is only an approach for calculational purposes. This means it is not something permanent. Meaning Quantum Field Theory can someday be replaced by others which doesn't involve the quantum field especially the matter field by second quantization. Do you agree with this analysis? Therefore when MySearch asked in the other thread "what is the field in QFT?". Well. I think the fields are just for certain calculational approach and is not something definite like spin and can be replaced someday. Do you agree?

I'm not sure what that would even mean. You're suggesting that quantum fields can be replaced by something else. How? Like when "action at a distance" (of the gravitational force) was replaced by the view that spacetime is a manifold with a metric to be determined from an equation. That gave us a completely different theory. Is that the sort of thing you're talking about? Or are you suggesting that there's a better way to state theories like QED, that may not involve fields? I very much doubt that there is, and even if there is, the fields would still be present in the theory.

Here's a quote from Steven Weinberg:

...it is very likely that any quantum theory that at sufficiently low energy and large distances looks Lorentz invariant and satisfies the cluster decomposition principle will also at sufficiently low energy look like a quantum field theory.​

It's from this transcript of one of his talks, but he's also mentioning this idea in his QFT book.

 

[waterfall]

I'm not sure what that would even mean. You're suggesting that quantum fields can be replaced by something else. How? Like when "action at a distance" (of the gravitational force) was replaced by the view that spacetime is a manifold with a metric to be determined from an equation. That gave us a completely different theory. Is that the sort of thing you're talking about? Or are you suggesting that there's a better way to state theories like QED, that may not involve fields? I very much doubt that there is, and even if there is, the fields would still be present in the theory.

Here's a quote from Steven Weinberg:

...it is very likely that any quantum theory that at sufficiently low energy and large distances looks Lorentz invariant and satisfies the cluster decomposition principle will also at sufficiently low energy look like a quantum field theory.​

It's from this transcript of one of his talks, but he's also mentioning this idea in his QFT book.

So quantum field will be with us forever. I heard QFT could be low energy limit of superstrings or something. So you mean there may be a larger theory but QFT will just be the classical limit of it?

I wonder what else besides Superstrings or M-Theory that can comprise the larger theory...

 

[juanrga]

I'm trying to understand the basics of convensional QFT versus QM. There are too many books about QM in the introductory level for layman but too rare for QFT. But the public needs to be adept about QFT too not just particle-wave duality, entanglement and other attractions in QM.

Let's start by a table or FAQ of some kind distinguishing QFT and QM. Maybe QFT is not so hard after all.

1.
QM uses Hilbert Space.
QFT uses Fock Space.

(Since Hilbert Space is not in physical 3D, then Fock Space is not in physical 3D either, it is in so called abstract configuration space.. therefore automatically quantum fields are not physical in convensional QFT, is this reasoning correct?)

2.
QM has position as observable.
QFT has position as operator (in other words, you can consider these as self-observing, isn't it)
How about momentum and spin? Are these observables or operators in QFT?

3.
QM uses no relativity.
QFT uses relativity in the sense of mass converting to energy and vice versa even if the speed is not near light (so the SR sense is more of E=mc^2 and not speed, correct?)

4.
QED, QCD, and EWT is an application of convensional QFT. In QED. It is natural to quantize the electromagnetic wave or field and produce the harmonic oscillators as photons. What's oscillating are magnetic field and electric field and displacement current via the Maxwell Equations. Steve Weinberg mentioned all particles are actual energy and momentum of the fields. But in electron, what is the equivalent of the electromagnetic field in QED that uses Maxwell Equations? What's oscillating in electron wave/field or the magnetic field/electric field counterpart of it?

(if you can add some basic FAQ of difference between QM and QFT, please add it so we can enable the millions of laymen in QM to understand QFT too in the basic level, thanks.)

Thanks.

1.
Fock Space is based in Hilbert space.

2.
QM has position as observable because has operator.
QFT has not position operator and position is not obdservable but a dummy unphysical parameter.

Momentum and spin are observables given by operators in QFT

3.
(Non-relativistic) QM uses no relativity.
(relativistic) QFT uses relativity in the sense of using a dummy version of special relativity, where x and t are not measurable.

4.
QED, QCD, and QWD are examples of QFT.

The equivalent of the electromagnetic field in QED for electrons (fermions) is the fermion field.

 

[waterfall]

1.
Fock Space is based in Hilbert space.

2.
QM has position as observable because has operator.
QFT has not position operator and position is not obdservable but a dummy unphysical parameter.

Momentum and spin are observables given by operators in QFT

3.
(Non-relativistic) QM uses no relativity.
(relativistic) QFT uses relativity in the sense of using a dummy version of special relativity, where x and t are not measurable.

4.
QED, QCD, and QWD are examples of QFT.

The equivalent of the electromagnetic field in QED for electrons (fermions) is the fermion field.

After days of discussions. I know all of them already. But I have new questions.

I heard it said that an electron around a proton or even a traveling single electron can be modeled by QFT. So how does one start to do that? I want to imagine the matter field of electron and proton and how they behave and also the matter field of the single traveling electron. I know QFT is appropriate for an "infinite numbers of particles". But again I heard it can be done for a single or two particles. How?

 

[mysearch]

I got the feeling that somehow QFT is only an approach for calculational purposes……Therefore when MySearch asked in the other thread "what is the field in QFT?". Well. I think the fields are just for certain calculational approach and is not something definite like spin and can be replaced someday.

Waterfall, my apologises for posting a general reply to one of your earlier posts, as I can see you are anxious to try to get specific answers to the following questions.

I heard it said that an electron around a proton or even a traveling single electron can be modeled by QFT. So how does one start to do that? I want to imagine the matter field of electron and proton and how they behave and also the matter field of the single traveling electron. I know QFT is appropriate for an "infinite numbers of particles". But again I heard it can be done for a single or two particles. How?

Whether this is entirely possible is unclear to me, if QFT is not a final theory, but I will continue track the thread with interest. However, one of the confusing things about QFT, at least to me, is that at one level it underwrites the standard particle model, which has amassed a huge amount of observational data suggesting that the assumptions of QFT must reflect some tangible aspect of quantum reality, but which I assume we ultimately describe in terms of a classical approximation, e.g. a particle. However, this said, the ‘reality’ of the developing quantum description still seems to be surrounded by much ambiguity, given the level of scientific, mathematical and philosophical conjecture, e.g.

Why Quantum Theory?
“The usual formulation of quantum theory is very obscure employing complex Hilbert spaces, Hermitean operators and so on. While many of us, as professional quantum theorists, have become very familiar with the theory, we should not mistake this familiarity for a sense that the formulation is physically reasonable. Quantum theory, when stripped of all its incidental structure, is simply a new type of probability theory.”

So What are the Fields in QFT? Here is a summary of some suggested descriptions from the thread referenced above.

….There are scalar, vector, and tensor quantum fields…..Quantum fields have energy and momentum, but are not "energy fields", but fermion fields, boson fields...The photon is the quanta of the EM quantum field. Each field and its quanta has different properties as charge, spin, mass...

Regarding fields they are modeled as a collection of harmonic oscillators. And if you ask what is oscillating? Then either you avoid to answer or you return to a particle concept. Moreover, the concept of field is only approximate. It is now generally accepted that QFT is only an effective theory that breaks down to higher energies. Field theory also breaks in other situations, and alternatives are under active research.

A particle is an object with determined properties assigned to it. An elementary particle is a microscopic non-composite object characterized by mass, spin, charge...Energy and position are not properties that define what a particle is. Moreover a particle does not need to be confined in a small volume of space. The term «matter wave» is a misnomer for me.

How do other references define/quantify fields? The first essentially appears to align with Juanrga’s breakdown of the ‘types’ of fields referenced in QFT.

Quote taken from p.41 summarised:
Particles with zero spin, such as pions and the famous Higgs boson, are known as scalars, and are governed by the Klein-Gordon equation. Particles with ½ spin, such as electrons, neutrinos, and quarks, and known as spinors, defined by the Dirac equation. And particles with spin 1, such as photons and the W’s and Z’s that carry the weak charge, and known as vectors discovered by Alexandru Proça. The Proça equation reduces, in the massless (photon) case, to Maxwell’s equations.

However, the tangibility of these fields then seems to recede in the following (1.8) paragraph on p.9
When the word “field” is used classically, it refers to an entity, like fluid wave amplitude, E, or B, that is spread out in space, i.e., has different values at different places. By that definition, the wave function of ordinary QM, or even the particle state in QFT, is a field. But, it is important to realize that in quantum terminology, the word “field” means an operator field, which is the solution to the wave equations, and which creates and destroys particle states. States (= particles = wave functions = kets) are not considered fields in that context.

So at one level, the idea of scalar, spinor and vector fields seems rooted in a mathematical description, although at another level the quantization of the EM field into photon particles almost seems tangible. Of course, one might still have to question the physicality of a photon in spacetime. For example, here are some further clarifications of the idea of a field in QFT taken from this thread:

…quantum field is a quantum 'quantity'. In the formalism of quantum physics, these are operators (or POVMs, which are a related but more complicated object). The 'actual' field IS the 'operator' field.

What I meant is, the operator-valued fields are the mathematical models that correspond in the quantum theory to the number-valued fields of the classical theory.

This seems a bit strange; what does it mean for something to be "in physical 3D" and what does it mean for something to be "physical?" I think you can make a strong case that at least the electromagnetic field is "physical"--it is fairly directly measurable. And the electromagnetic field, properly treated, is a quantum field…..
A quantum field is really a set of operators, one at each point in spacetime; i.e., an infinite set of operators, each "labelled" by a spacetime position. ...
In QFT we define an "electron field" whose quantized oscillations are electron particles. The electron field is a bit of a weird thing, though. For instance it is not directly observable.

As a somewhat off-the-cuff thought, has anybody ever attempted to quantify the nature of energy as a scalar quantity and its apparent ability to move in spacetime in terms of some fundamental ability of spacetime to be distorted? In part, it seems that that general relativity alludes to the idea of curved spacetime, such that we might re-interpret John Wheeler’s original quote:

“Matter tells spacetime how to curve, and spacetime tells matter how to move!”. ->
“Energy tells spacetime how to curve, and spacetime tells energy how to move?”

Please accept this as a question, not as a proposal, but could spacetime itself be the basis of the field?

 

[juanrga]

After days of discussions. I know all of them already. But I have new questions.

I heard it said that an electron around a proton or even a traveling single electron can be modeled by QFT. So how does one start to do that? I want to imagine the matter field of electron and proton and how they behave and also the matter field of the single traveling electron. I know QFT is appropriate for an "infinite numbers of particles". But again I heard it can be done for a single or two particles. How?

Only some aspects of the single electron or of the electron around a proton can be studied in QFT. There is not such thing as «the matter field of electron and proton» but a field for the electron and other for the proton.

It is not right that QFT is appropriate for an "infinite numbers of particles", because those systems are plagued with infinities, which have to be regularized and renormalized. Such techniques are not general.

 

[waterfall]

mysearch, in M.Y. Han book. It is mentioned that the gauge symmetry craze in the 1970s have physicists hooked on QFT because of the electromagnetism U(1) which clued them to electroweak U(1)xSU(2) and strong force is SU(3), this third phase is called the (Lagrangian) gauge field theory. This is what made them forgive QFT having non-interacting fields.. because they think gauge theory can somehow save the day. But I wonder if gauge theory can also be hold on without the path of QFT exactly (does anyone know the answer?). The M.Y. Han book can give you a bird eye view of QFT. If you have other interesting QFT book recommendation which you have read or encountered, let me know. Thanks.

 

[waterfall]

To people who only participate in this quantum forum. I learned from M.Y. Han book that there are 3 phases of development of quantum field theory and how they deal with non-interacting fields. I'll summarize it.

First phase (Early 1950s) - Langrangian Field Theory - based on canonical quantization, success in QED followed by non-expandability in the case of strong nuclear force and by non-renomalizability in the case of weak nuclear force.

Second phase (1950s-1960s) - Axiomatic QFT - for example S-Matrix theories and other axiomatic approaches, however they did not bring solutions to quantum field theories any closer than the Lagrangian field theories.

Third phase (1970s) - (Lagrangian) gauge field theory - ongoing

My question is. Can you make use of Gauge Theory without using Quantum Field Theory? Or the two completely related? But noether theorem can be applied to Newtonian physics so can the gauge symmetry concept of electromagnetism U(1), electroweak U(1)xSU(2), Strong SU(3) can be developed without using the concept of quantum field theory?

 

[Fredrik]

I don't think the idea of gauge fields is useful in classical field theory. Take electrodynamics for example. How do you modify the classical theory of an electromagnetic field in Minkowski spacetime to make it gauge invariant? By introducing the electron/positron field, which is a spin-1/2 field. I think the gauge fields are always fermionic (half-integer spin) and that this is what makes them useless in a classical context.

Edit: This is obviously wrong. I realized that after seeing atyy's post. See my correction in post #54.

 

[waterfall]

I don't think the idea of gauge fields is useful in classical field theory. Take electrodynamics for example. How do you modify the classical theory of an electromagnetic field in Minkowski spacetime to make it gauge invariant? By introducing the electron/positron field, which is a spin-1/2 field. I think the gauge fields are always fermionic (half-integer spin) and that this is what makes them useless in a classical context.

Classical electrodynamics is already automatically lorentz invavariant. In fact Einstein built the SR from following it.

So I guess gauge invariance is another issue. Are you sure spin 0 and spin 2 can't be properties of gauge invariance but only spin 1/2? How come?

Btw.. in QED.. do they analyze the electric field as coulomb potential or only as virtual particles... like every analysis in QED involves perturbation of particles?

 

[Fredrik]

So I guess gauge invariance is another issue.

Yes. A gauge transformation isn't a coordinate transformation.

Are you sure spin 0 and spin 2 can't be properties of gauge invariance but only spin 1/2? How come?

No, I'm not sure because the only gauge theory I've studied is QED, and it was a long time ago. Now that you mention it, gravitons have spin 2. Not sure what that means though. There are a few threads here where the question of whether gravity is a gauge theory is debated. I think the conclusion was that it's not a gauge theory in the traditional sense, but the answer still depends on what exactly you mean by a gauge theory. I still think that what I said is correct, but if someone tells you that I'm not and they sound like they know what they're talking about, they're probably right.

Btw.. in QED.. do they analyze the electric field as coulomb potential or only as virtual particles... like every analysis in QED involves perturbation of particles?

I think it can be treated as a potential in approximate calculations, but as I said, it was a long time ago.

 

[Fredrik]

My question is. Can you make use of Gauge Theory without using Quantum Field Theory? Or the two completely related? But noether theorem can be applied to Newtonian physics so can the gauge symmetry concept of electromagnetism U(1), electroweak U(1)xSU(2), Strong SU(3) can be developed without using the concept of quantum field theory?

I didn't answer the last part. Yes, it can be developed in an entirely classical setting, using fiber bundle theory. The mathematics is pretty heavy. The classical theories that are found this way are however pretty useless until they are quantized in one way or another.

 

[atyy]

The electromagnetic field is a gauge field when potentials are used, as they are QFT.

The more common definition of a gauge field just means that several different ways of naming the field are physically equivalent. So electric potential in circuit theory has a gauge invariance in this sense - it is only potential difference that is physical, the potential itself can be shifted arbitrarily. In the same sense, the diffeomorphism invariance is a gauge invariance - metrics that are related by diffeomorphisms are physically equivalent. This is why you will see the term "de Donder gauge" with reference to classical general relativity.

There is a second different definition of a gauge field as the connection on a bundle, and gravity is not a gauge field in this sense.

 

[Fredrik]

Ah, I was confused about the most important detail. I probably shouldn't be posting this late at night. I remembered that QED is found by taking one theory and adding another field to make the theory gauge invariant. But I was thinking that this process adds the Dirac field to electromagnetism, when in fact it's the other way round. You start with the Lagrangian for a non-interacting Dirac field, note that it's not gauge invariant, and add a vector (spin-1) field with special properties to get a theory that is gauge invariant. This vector field is the electromagnetic 4-potential.

 

[atyy]

Oh yes, there's another interesting point. Actually there is more than one way to make the Dirac and EM fields interact while having EM gauge invariance. The usual "gauge principle" is maybe more informatively called "minimal coupling" - just as the "equivalence principle" of GR is really a "minimal coupling" of matter and metric.

 

[waterfall]

Ah, I was confused about the most important detail. I probably shouldn't be posting this late at night. I remembered that QED is found by taking one theory and adding another field to make the theory gauge invariant. But I was thinking that this process adds the Dirac field to electromagnetism, when in fact it's the other way round. You start with the Lagrangian for a non-interacting Dirac field, note that it's not gauge invariant, and add a vector (spin-1) field with special properties to get a theory that is gauge invariant. This vector field is the electromagnetic 4-potential.

U(1) gauge invariance is supposed to be that of electromagnetism. But in QED, electrons or spin 1/2 are involved because it is supposed to be an interaction between light and matter. So how come they sort of ignored the spin 1/2 of matter waves and just focus on the photon spin 1? Why not spin 1 + spin 1/2 which is not gauge invariant?

 

[sheaf]

Oh yes, there's another interesting point. Actually there is more than one way to make the Dirac and EM fields interact while having EM gauge invariance. The usual "gauge principle" is maybe more informatively called "minimal coupling" - just as the "equivalence principle" of GR is really a "minimal coupling" of matter and metric.

Oh that sounds interesting. How do you build the Lagrangian without resorting to minimal coupling?

 

[Demystifier]

Are you saying not all physicists with Ph.D. are experts in QFT? I thought they all wer.

That shows how little you know about physics in general. Before styding QFT, I would recommend you to start with more elementary stuff, such as classical mechanics, classical field theory, classical electrodynamics, and elementary quantum mechanics, before attempting to refute QFT theories published in peer review journals.

 

[Fredrik]

U(1) gauge invariance is supposed to be that of electromagnetism. But in QED, electrons or spin 1/2 are involved because it is supposed to be an interaction between light and matter. So how come they sort of ignored the spin 1/2 of matter waves and just focus on the photon spin 1? Why not spin 1 + spin 1/2 which is not gauge invariant?

What makes you think something has been ignored? I don't follow you here. What I said is that if you take the theory with just electrons and positrons that don't interact with anything including themselves, it's not U(1) invariant, but if you add an interaction term involving the electromagnetic 4-potential (i.e. photons), you get a U(1) invariant theory.

Oh that sounds interesting. How do you build the Lagrangian without resorting to minimal coupling?

You can add more gauge invariant terms (products of fields and field derivatives with more factors), in addition to the simplest one. The problem, in the case of QED at least, is that the simplest possibility is the only one that gives us a renormalizable theory. I think the theory with all gauge invariant terms added would be the most accurate, if we could find a way to do calculations with it, but no one cares, since the contribution from the non-renormalizable terms to low-energy processes is negligible anyway, and since the predictions made by the renormalizable theory are accurate enough that experiments with current technology can't find anything wrong with the theory.

Maybe you could also add additional gauge fields. I'm not sure. I think in that case, it wouldn't be a U(1) gauge theory anymore.

 

[sheaf]

You can add more gauge invariant terms (products of fields and field derivatives with more factors), in addition to the simplest one. The problem, in the case of QED at least, is that the simplest possibility is the only one that gives us a renormalizable theory. I think the theory with all gauge invariant terms added would be the most accurate, if we could find a way to do calculations with it, but no one cares, since the contribution from the non-renormalizable terms to low-energy processes is negligible anyway, and since the predictions made by the renormalizable theory are accurate enough that experiments with current technology can't find anything wrong with the theory.

Maybe you could also add additional gauge fields. I'm not sure. I think in that case, it wouldn't be a U(1) gauge theory anymore.

Ah thanks, I often wondered what the "minimal" in "minimal coupling" was referring to!